Harmonic analysis on solvable Lie groups associated with constructions of induced representations

与诱导表示构造相关的可解李群的调和分析

• 批准号: 15540171
• 负责人: INOUE Junko
• 金额: $ 1.41万
• 依托单位: Tottori University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 2003
• 资助国家: 日本
• 起止时间: 2003 至 2005
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/en/grant/KAKENHI-PROJECT-15540171/
• 关键词: solvable Lie group; induced representation; non-commutative harmonic analysis; orbit method; coadjoint orbit; 複素解析的誘導表現; polarization

1.According to the orbit method, an irreducible unitary representation of an exponential solvable Lie group is realized as an induced representation from a unitary character of a connected subgroup on a space of L^2-functions of the homogeneous space diffeomorphic with R^n. When the group is nilpotent, the space of C^∞ vectors coincides with the space of Schwartz functions of R^n, and it fits in with a description of the Fourier transform for the group. However, when the group is general exponential, the space of C^∞ vectors does not have such simple descriptions. As a collaboration with J.Ludwig (University of Metz, France), we investigated C^∞ vectors for irreducible representations of exponential groups G, and obtained the following results : Taking a nilpotent ideal which includes the derived ideal of the Lie algebra of G, and choosing a real polarization adapted to the ideal, we realize the representation on a space of L^2-functions on the homogeneous space. We treat a subspace consisting of functions with some rapidly decreasing property associated with the nilpotent ideal, and show that it can be embedded as the space of C^∞ vectors for an irreducible representation of a group containing G. Using the space, we also describe a certain space of functions included in the image of Fourier transforms of L^1-functions on G of finite ranks. Furthermore, specific descriptions of the space of C^∞ vectors for some special classes of representations are obtained.2.I investigated generalized holomorphically induced representations, and described associated semiinvariant vectors for some examples of low dimensional solvable Lie groups. Their descriptions depend on particular group structures.

1.根据轨道法的符合，可溶解的谎言组的不可还原的单一表示是从连接子组的单位特征的诱导表示，这是在均匀空间的l^2函数的空间上与r^n的均匀空间差异。当组为nilpotent时，c^∞矢量的空间与r^n的Schwartz函数的空间一致，并且与该组的傅立叶变换的描述相符。但是，当组是一般指数时，C^∞向量的空间没有那么简单的描述。 As a collaboration with J.Ludwig (University of Metz, France), we investigated C^∞ vectors for irreducible representations of exponential groups G, and obtained the following results: Taking a nilpotent ideal which includes the derived ideal of the Lie algebra of G, and choosing a real polarization adapted to the ideal, we realize the representation on a space of L^2-functions on the homogeneous space. We treat a subspace consisting of functions with some rapidly decreasing property associated with the nilpotent ideal, and show that it can be embedded as the space of C^∞ vectors for an irreducible representation of a group containing G. Using the space, we also describe a certain space of functions included in the image of Fourier transforms of L^1-functions on G of finite ranks.此外，获得了某些特殊类别的C^∞矢量空间的特定描述。2.i研究了通常荷晶诱导的表示，并描述了与一些低维溶解层组的一些示例相关的半变量矢量。他们的描述取决于特定的组结构。